Invariants
| Level: | $16$ | $\SL_2$-level: | $16$ | ||||
| Index: | $96$ | $\PSL_2$-index: | $48$ | ||||
| Genus: | $0 = 1 + \frac{ 48 }{12} - \frac{ 0 }{4} - \frac{ 0 }{3} - \frac{ 10 }{2}$ | ||||||
| Cusps: | $10$ (of which $2$ are rational) | Cusp widths | $2^{8}\cdot16^{2}$ | Cusp orbits | $1^{2}\cdot2^{4}$ | ||
| Elliptic points: | $0$ of order $2$ and $0$ of order $3$ | ||||||
| $\Q$-gonality: | $1$ | ||||||
| $\overline{\Q}$-gonality: | $1$ | ||||||
| Rational cusps: | $2$ | ||||||
| Rational CM points: | none |
Other labels
| Cummins and Pauli (CP) label: | 16G0 |
| Rouse and Zureick-Brown (RZB) label: | X217e |
| Rouse, Sutherland, and Zureick-Brown (RSZB) label: | 16.96.0.306 |
Level structure
| $\GL_2(\Z/16\Z)$-generators: | $\begin{bmatrix}1&7\\0&15\end{bmatrix}$, $\begin{bmatrix}5&0\\8&15\end{bmatrix}$, $\begin{bmatrix}11&7\\8&5\end{bmatrix}$ |
| $\GL_2(\Z/16\Z)$-subgroup: | $C_4^2.\SD_{16}$ |
| Contains $-I$: | no $\quad$ (see 16.48.0.e.1 for the level structure with $-I$) |
| Cyclic 16-isogeny field degree: | $2$ |
| Cyclic 16-torsion field degree: | $8$ |
| Full 16-torsion field degree: | $256$ |
Models
This modular curve is isomorphic to $\mathbb{P}^1$.
Rational points
This modular curve has infinitely many rational points, including 7 stored non-cuspidal points.
Maps to other modular curves
$j$-invariant map of degree 48 to the modular curve $X(1)$ :
| $\displaystyle j$ | $=$ | $\displaystyle \frac{1}{2}\cdot\frac{(2x-y)^{48}(24832x^{8}+33792x^{7}y-164096x^{6}y^{2}-119040x^{5}y^{3}+166496x^{4}y^{4}-6720x^{3}y^{5}-22288x^{2}y^{6}+4752xy^{7}-191y^{8})^{3}(48896x^{8}-304128x^{7}y+356608x^{6}y^{2}+26880x^{5}y^{3}-166496x^{4}y^{4}+29760x^{3}y^{5}+10256x^{2}y^{6}-528xy^{7}-97y^{8})^{3}}{(2x-y)^{50}(2x+y)^{2}(4x^{2}-12xy+y^{2})^{2}(4x^{2}+4xy-7y^{2})^{2}(12x^{2}-4xy+3y^{2})^{16}(28x^{2}-4xy-y^{2})^{2}}$ |
Modular covers
This modular curve minimally covers the modular curves listed below.
| Covered curve | Level | Index | Degree | Genus | Rank |
|---|---|---|---|---|---|
| 8.48.0-8.k.1.4 | $8$ | $2$ | $2$ | $0$ | $0$ |
| 16.48.0-8.k.1.2 | $16$ | $2$ | $2$ | $0$ | $0$ |
| 16.48.0-16.e.1.12 | $16$ | $2$ | $2$ | $0$ | $0$ |
| 16.48.0-16.e.1.13 | $16$ | $2$ | $2$ | $0$ | $0$ |
| 16.48.0-16.e.2.4 | $16$ | $2$ | $2$ | $0$ | $0$ |
| 16.48.0-16.e.2.14 | $16$ | $2$ | $2$ | $0$ | $0$ |
This modular curve is minimally covered by the modular curves in the database listed below.